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IntroductionJohn F. McClelland1,2, Roger W. Jones2, and Stanley J. Bajic2
1MTEC Photoacoustics, Inc. PO Box 1095
Ames, IA 50014
February, 2002
This is a print of a chapter that appears in Handbook of Vibrational Spectroscopy
edited by John M. Chalmers and Peter R. Griffiths and published by John Wiley & Sons, Ltd.
The desired result with all Fourier transform infrared (FT-IR) sampling techniques is to obtain an absorbance spectrum of the sample as quickly and easily as possible. In many cases, however, direct analysis of “as received” samples by transmission or reflection methods is not practical because the sample either transmits inadequate light to measure or it lacks suitable surface or particle size conditions for reflectance spectroscopies. In other cases, reflectance spectroscopies may not probe deeply enough into the sample to yield the desired information. Photoacoustic spectroscopy (PAS)1-3 is unique as a sampling technique, because it does not require that the sample be transmitting, has low sensitivity to surface condition, and can probe over a range of selectable sampling depths from several micrometers to more than 100 µm. PAS has these capabilities because it directly measures infrared (IR) absorption by sensing absorption-induced heating of the sample within an experimentally controllable sampling depth below the sample’s surface. Heat deposited within this depth transfers to the surrounding gas at the sample surface, producing a thermal-expansion- driven pressurization in the gas, known as the PAS signal, which is detected by a microphone. The magnitude of the PAS signal varies linearly with increasing absorptivity, concentration or sampling depth until at high values of their product a gradual roll off in sensitivity (saturation) occurs. The phase of the PAS signal corresponds to the time delay associated with heat transfer within the sample. These signal components are described in detail in the next section. PAS signal generation is initiated when the FT-IR beam, which oscillates in intensity, is absorbed by the sample resulting in the absorption-induced heating in the sample and oscillation of the sample temperature. The temperature oscillations occurring in each light-absorbing layer within the sample launch propagating temperature waves called thermal-waves, which decay strongly as they propagate through the sample. It is this thermal-wave decay process that defines the layer thickness, or sampling depth, from which spectral information is obtained in an FT-IR PAS analysis. The sampling depth can be increased by decreasing, via FT-IR computer control, the IR beam modulation frequency imposed by the interferometer. The lower modulation frequency allows a longer time for thermal-waves to propagate from deeper within the sample into the gas. As the sampling depth increases, the saturation of strong bands in PAS spectra increases just as it does in absorption spectra measured by transmission as sample thickness increases. The discovery of the photoacoustic effect by Alexander Graham Bell in 1880 marked the beginning of the development of the technique as a useful spectroscopic method.4 Development was hampered, however, by the weak acoustic signals that must be measured due to the very high thermal-wave reflection coefficient at the sample-to-gas interface. A high fraction of the thermal-wave amplitude is reflected back into the sample and is not detected, leading to signal-to-noise problems. Signal saturation also was a problem in the initial efforts to apply the technique in the ultraviolet and visible spectral regions. Operation in the near- and mid-infrared spectral regions, made practical
with the multiplexing capability of FT-IR systems and the higher sensitivity of photoacoustic detectors, has been a major area of success for the PAS technique. These spectral regions are rich in chemical information, and modern search and chemometric software allow qualitative and quantitative results to be readily obtained from PAS spectra in the presence of the more modest saturation effects found in these spectral regions. At this time, FT-IR PAS is a broad field of research that continues to develop in the areas of instrumentation, applications, and data analysis. This article will be restricted to primarily FT-IR PAS of solid samples and to a much lesser degree of liquids.
The article Transient Infrared Spectroscopy for on-line analyses may also be of interest to readers because transient infrared spectroscopy (TIRS) is also a thermal-wave based technique that has similar capabilities to PAS but operates on moving samples.
Photoacoustic Signal Generation, Processing and Interpretation The photoacoustic signal contains information on the sample’s absorption
spectrum and on the depth below the sample’s surface from which the signal evolves, allowing materials with layered or gradient compositions to be studied. Photoacoustic signal generation can be modeled5,6 using the heat equation7 and assuming a one- dimensional heat flow within the sample and adjacent gas atmosphere that is in the direction opposite to that of the light beam. The most instructive model for general purposes also assumes an optically and thermally homogeneous slab sample geometry which is thick on the scale of the thermal-wave decay length with the rear sample face thermally grounded and optically nonreflective. The model is shown schematically in Figure 1.
Figure 1. One-dimensional signal generation schematic showing the decay length, L, for thermal-waves and the optical decay lengths for lower (α1) and higher (α2) values of absorption coefficient. As α increases, more of the absorption occurs in the region near the sample’s surface that is active in signal generation. Reproduced from Reference 3.
The FT-IR interferometer modulates the intensity of the infrared beam that is incident on the sample. The beam is partially reflected (RI0) at the front face of the sample but this reflection is ignored in the simple model. The beam then decays
exponentially with an absorption coefficient, α(ν∼ ), as it propagates within the sample.
The wavenumber of the infrared radiation is denoted by ν∼ . In most cases all of the absorbed radiation is converted into heat, causing the temperature of each absorbing layer to oscillate at the beam modulation frequency with an amplitude proportional to the amount of light absorbed in it. Each of these layers becomes a source for launching propagating temperature oscillations called thermal waves.
Thermal waves have three important properties affecting photoacoustic signal generation. They have a short decay length called the thermal diffusion depth or thermal wave decay length, L, given by equation (1):
L = (D/πf)1/2 (1)
where D and f denote the sample’s thermal diffusivity and the infrared beam modulation frequency, respectively. Thermal waves decay to 37% (i.e. 1/e) of their original amplitude over a distance of L. If the decay of thermal waves did not define an active signal-generation layer that is smaller than the optical decay length, photoacoustic spectra of opaque samples would be just as hopelessly saturated and impractical for measurement as transmission spectra are for such samples. Fortunately, as long as the thin layer is partially transmitting, the photoacoustic signal increases with absorption coefficient and spectra can be readily measured by PAS, regardless of sample thickness. After the thermal waves are launched, those that propagate to the front face of the sample contribute to the PAS signal, but most of their amplitude is not detected because it is reflected back into the sample and decays. The strong back reflection of thermal waves in the solid is the second important property of thermal waves in photoacoustic signal generation, but it is not as fortuitous for the signal generation process as is their short decay length. In fact, if the high back reflection were not present, photoacoustic signals would have significantly higher amplitudes and signal-to-noise ratios. The small thermal-wave amplitude that does transmit into the gas results in thermal expansion and a pressure oscillation in the gas, which increases with
α(ν∼ ) and that is detected as an acoustic signal containing both phase and magnitude information by a sensitive microphone. The phase of the PAS signal is equal to the phase lag between the signal and the waveform of the IR beam that excites it. The lag is caused by the finite propagation time of thermal waves during signal generation. This is the third important property of thermal waves in signal generation and it results in the phase angle being a measure of the depth from which the signal evolves within the sample. The maximum phase angle that can be measured is one cycle, or 360°, corresponding to a maximum depth of 2πL on the length scale. In many instances, PAS data are analyzed in the form of magnitude spectra. These spectra are commonly used for qualitative and quantitative analysis of
homogeneous samples. Nonhomogeneous samples that have been either homogenized by size reduction or sampled multiple times to assure good sampling statistics, are also suitable for analysis using magnitude spectra. Magnitude spectra measured at different modulation frequencies are used to study samples having compositional variations as a function of depth often in combination with phase spectra. PAS signal dependence on α is nonlinear at the extremes of low and high values of α. The PAS magnitude signal experiences a background signal “floor” at very low values of α (typically less than 1 cm-1), due to the absorption of light by contamination on the sample chamber walls and on the sample itself, and to an acoustic-piston6 sample response. Above this floor, the signal goes through a linear range with increasing α as predicted by the simple model until at a value of α, denoted by αo the signal experiences an onset of signal saturation and starts to loose sensitivity to increasing values of α. Full saturation occurs at a higher value of α, denoted by αf,, where the PAS signal no longer senses increases in absorption coefficient. Table 1 gives the absorption coefficient values for different modulation frequencies estimated from the simple model for the onset and full saturation conditions, assuming a homogeneous sample. Note that as the modulation frequency increases, αo and αf shift to higher absorption coefficient values due to a thinning of the active layer generating the signal. Another important point is that the absorption coefficient continues to increase for over two orders of magnitude above the onset of saturation before the signal loses all sensitivity to α at αf. The simple model can be used to make useful connections between the sample’s optical and thermal properties resulting in approximate rule of thumb values for αo, αf, and αll as a function of L: αo ≅ 1/10L (2) αf ≅ 20/L (3)
αll ≅ 100/L (4) αll of Equation 4 is referred to as the linearization limit and it marks the practical upper limit that can be reached when photoacoustic magnitude and phase data are combined to calculate so-called linearized magnitude spectra.3,8-10 Linearized PAS spectra vary linearly with α to approximately three orders of magnitude above the onset of saturation observed for purely magnitude spectra. The linearization process involves measuring single beam spectra for the sample and for a glassy carbon reference, shifting the interferograms, if necessary, so that they have the same centerburst retardations, and transforming the interferograms using the same phase correction to obtain the real (R) and imaginary (I) components. The linearized spectrum, Sl, is then calculated from10
Sl = (SR 2 + SI
2)/1.414 (SIRR-SRRI) (5) where the subscripts R and I denote the real and imaginary components of the sample (S)
and reference (R) spectra. Explicit ν∼ dependence has been omitted in Equation (5).
Table 1. Rule of thumb absorption coefficient values (cm-1) derived from the simple model that characterizes PAS signal generation in polymers assuming D = 10-3 cm2/s. 10 Hz 100 Hz 1000 Hz 10,000 Hz
αo (onset of saturation) 18 56 180 560
αf (full saturation) 3300 1.1 · 104 3.3 · 104 1.1 · 105
αll (linearization limit) 2 · 104 6.3 · 104 2 · 105 6.3 · 105
Linearization of magnitude spectra has two primary advantages. The most important is that the sampling depth in effect is reduced by approximately a factor of 3 going from pure magnitude to linearized magnitude spectra. This results in the emphasis of spectral features due to thin surface layers with higher surface specificity than can be obtained from a purely magnitude spectrum taken at the highest FT-IR mirror velocity available. The second advantage is, of course, the reduction or removal of saturation in strong absorbance bands, which may be valuable in quantitative analyses. In many cases, however, chemometric programs tolerate significant amounts of saturation in spectra very well and produce excellent quantitative data, making linearization unnecessary for this purpose. Another issue related to signal saturation occurs when sampling depth is varied to investigate samples of depth-varying composition. As the modulation frequency is increased to produce shallower sampling, the magnitude of spectral bands change for two reasons: (i) bands associated with species concentrated closer to the surface increase relative to bulk bands and vice versa; and (ii) bands that are saturated also increase relative to weaker bands due to reduction in the level of saturation since, in effect, a thinner sample is being analyzed. These two effects are easily observed if the two spectra, taken at different modulation frequencies, are scaled so that a weak matrix band, which has no saturation in either spectrum, is of constant amplitude. This has customarily been the practice in order to put the spectra on as common a scale as possible, but it leaves the problem of the separation of band changes due to composition versus saturation unresolved. The simple signal-generation model, however, provides a basis for converting a low-modulation-frequency spectrum into a spectrum with nearly the same degree of saturation as a higher-frequency spectrum. This PAS saturation compensation approach essentially removes the saturation differences while retaining any differences related to sample structure. The method uses only magnitude spectra and requires no phase information.
The conversion process is as follows. For a thermally thick sample with a negligible thermal expansion signal contribution and ignoring reflectance, Equation (41) of McDonald and Wetsel6 gives the photoacoustic signal, , as equation (6): )~(νS
)~( 22
)~( 0
gpssg σ
να ρ
γ π
ν (6)
where j = (-1)1/2, f is the modulation frequency, I0 is the incident light intensity, T0 and P0 are the ambient temperature and pressure, lg and γ are the thickness and heat-capacity ratio of the gas, α is the absorption coefficient of the sample, ρ)~(ν s and Cps are the density and heat capacity of the sample, g = (κgσg)/(κsσs), σg = (j2πf / Dg)1/2, σs = (j2πf / Ds)1/2, Dg and Ds are the thermal diffusivities of the gas and sample, κg and κs are the thermal conductivities of the gas and sample, and . For two spectra taken at modulation frequencies f and N f, the ratio of their magnitudes after normalization, Q , can be derived from Equation (7):
s/)~( σνα=r )~(ν
)~( ν ν
ν (7)
where r is evaluated at frequency f, and it is assumed that the reference used for normalization has the 1/f frequency dependence typical of a strong absorber. Of course, in nonphase-modulation FT-IR PAS the modulation frequency is not constant within a spectrum, but depends on wavenumber. Nevertheless, Equation (7) still applies because the modulation frequencies of the two spectra differ by N at each wavenumber when the scanning speeds of the two spectra differ by N. Multiplying a spectrum at one scanning speed by Q would convert it into a spectrum at N times the scanning speed, but because r is a sample-dependent quantity, Equation (7) by itself cannot be applied without knowing the physical properties of the sample. To produce a universal, sample-independent equation, r must be eliminated. This can be done by putting the low-frequency spectrum on a scale where the maximum possible signal is one. This scaled magnitude, S , can be derived from Equation (8):
Combining Equations (7) and (8) allows Q and to be given in terms of one another, independent of sample-specific quantities as shown in equations (9) and (10):
)~(ν )~(νscS
ν (10)
Figure 2 shows the relationship between and for selected frequency ratios. )~(νQ )~(νscS
0 0.2 0.4 0.6 0.8 1 0.2
N = 4
N = 8
N = 16
Ssc ν~
Q ν~
Figure 2. The ratio of the magnitudes of two normalized spectra whose scanning speeds differ by a ratio, N, as a function of the magnitude of the low-speed spectrum scaled to a maximum of 1. How are Equations (9) and (10) used to convert a low-scanning-speed spectrum into a high-scanning-speed equivalent? First, spectra at the two scanning speeds are acquired, such as the spectra in Figure 3 for a poly(methyl methacrylate) (PMMA) disk. The dashed- line spectrum was acquired at 0.158 cm/s (retardation change velocity) and the solid-line spectrum was acquired at 0.632 cm/s, so they differ by a speed ratio of 4; N = 4. Next, one peak in the spectra is chosen as a guide. This peak must arise solely from a homogeneously distributed component in a gradient sample or be a strong absorption in the first (outer) layer of a layered sample) so that it follows the homogeneous-sample behavior of the theory. Also, it is best if the peak is neither very weak, which could decrease the accuracy of the conversion, nor very strong, which could be approaching full saturation. The value of for that peak is determined from the peak's heights in the two spectra, and then for the peak is calculated from Q using Equation (10). The peak at 1153 cm
)~(νQ )~ν(scS
-1 can be the guide for the Figure 3 spectra; it has a Q value of 0.856. As Figure 2 illustrates, a value of 0.856 for Q means is 0.832 when N is four. The value for the guide peak fixes the scale for every point in the low-speed spectrum. For example, the 1450 cm
)~(ν )~(ν (scS
S )~(νscS )~ν
-1 peak in Figure 3 is seven-tenths the size of the guide peak (in the low-speed
spectrum). That means its value is 0.70(0.832) = 0.582. This scaling is done for the whole low-speed spectrum so that the value for every data point in the spectrum is known. Equation (9) is then used to determine what should be for every point in the spectrum. Multiplying each point in the low-speed spectrum by its value for converts the low-speed spectrum into the high-speed-spectrum equivalent. For the example peak at 1450 cm
)~(νscS )~(νscS
)-1, Figure 2 shows that its value of 0.582 corresponds to Q = 0.706, so in the conversion of the low-speed spectrum to the high-speed equivalent, the data point at the peak of the 1450 cm
-1 band is multiplied by 0.706. When this is done for every point in the low-speed spectrum, the spectrum shown by the dotted line in Figure 3 results. Because the PMMA sample is homogeneous and thermally thick, the converted spectrum should be identical to the true high-speed spectrum. They are very similar, if not quite identical. The difference between the two is shown by the heavy solid line near zero in Figure 3. This residual spectrum is not exactly zero because the simple-model theory of Equation (6) neglects various second-order effects. The most prominent error is the derivative-like shape of the residual spectrum at many peak locations. This comes from the characteristic change in sample reflectivity near absorption peaks, which is not included in the theory.
Q(ν) = 0.856~
Q(ν) = 0.706~
Figure 3. Photoacoustic magnitude spectra for a homogeneous PMMA sample, where the dashed line is the low frequency spectrum before saturation compensation and the dotted curve is after compensation. The lower trace is the high frequency spectrum minus the compensated…